#6386341

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,33)(24,37)(25,32)(26,30)(27,31)(28,36)(29,38)(34,48)(35,43)(39,46)(40,45)(41,44)(42,49)(47,51)(50,52);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(21,23)(22,28)(24,34)(25,40)(26,36)(27,35)(29,42)(30,33)(31,45)(32,43)(37,52)(38,51)(39,50)(41,49)(44,47)(46,48);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,39)(22,29,30,44)(23,34,36,50)(25,38,27,41)(28,42,33,47)(31,37,32,46)(35,49,40,51)(43,48,45,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(22,30)(28,33)(29,44)(31,32)(37,46)(42,47)(43,45)(48,52);(6,14)(11,15)(12,13)(19,20)(21,25,26,27)(22,31,30,32)(23,35,36,40)(24,38,39,41)(28,43,33,45)(29,37,44,46)(34,49,50,51)(42,48,47,52)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293537);